A solution to the problem of (A,B)-invariance for series
Résumé
The aim of this article is to study a standard problem of control theory, the (A,B)-invariance problem, which amounts to computing a maximal element X subject to conditions of the form AX less-than-or-equals X+B and X less-than-or-equals K. We give a solution to the problem in the framework of formal series over particular complete idempotent semirings. Over finite idempotent semirings, we show that, under the assumption that B and K are recognizable series, the maximal solution exists and is also recognizable. We obtain a similar result for the infinite tropical semiring, with additional hypothesis that the series A is a language, but the notion of recognizable series has to be extended to the weaker notion of pseudo-recognizable series.